Unruh effect

The Unruh effect (or sometimes Fulling–Davies–Unruh effect) is the prediction that an accelerating observer will observe blackbody radiation where an inertial observer would observe none. In other words, the background appears to be warm from an accelerating reference frame; in layman's terms, a thermometer waved around in empty space, subtracting any other contribution to its temperature, will record a non-zero temperature. For a uniformly accelerating observer, the ground state of an inertial observer is seen as in thermodynamic equilibrium with a non-zero temperature.

The Unruh effect was first described by Stephen Fulling in 1973, Paul Davies in 1975 and W. G. Unruh in 1976.[1][2][3] It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are disputed. There is also some doubt about whether the Unruh effect implies the existence of Unruh radiation.

where a is the local acceleration, kB is the Boltzmann constant, ħ is the reduced Planck constant, and c is the speed of light. Thus, for example, a proper acceleration of 7020247000000000000♠2.47×1020 m·s-2 corresponds approximately to a temperature of 7000100000000000000♠1 K. Conversely, an acceleration of 7000100000000000000♠1 m·s-2 corresponds to a temperature of 6979405999999999999♠4.06×10−21 K.

The Unruh temperature has the same form as the Hawking temperatureTH = ħg/2πckB of a black hole, which was derived (by Stephen Hawking) independently around the same time. It is, therefore, sometimes called the Hawking–Unruh temperature.[5]

Unruh demonstrated theoretically that the notion of vacuum depends on the path of the observer through spacetime. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles in thermal equilibrium—a warm gas.[6]

Although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word vacuum is interpreted in a specific way.

In modern terms, the concept of "vacuum" is not the same as "empty space": Space is filled with the quantized fields that make up the universe. Vacuum is simply the lowest possibleenergy state of these fields.

The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate. According to special relativity, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system. Hence, the observers will see different quantum states and thus different vacua.

In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation relations. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices.

An accelerating observer will perceive an apparent event horizon forming (see Rindler spacetime). The existence of Unruh radiation could be linked to this apparent event horizon, putting it in the same conceptual framework as Hawking radiation. On the other hand, the theory of the Unruh effect explains that the definition of what constitutes a "particle" depends on the state of motion of the observer.

The Rindler spacetime has a horizon, and locally any non-extremal black hole horizon is Rindler. So the Rindler spacetime gives the local properties of black holes and cosmological horizons. The Unruh effect would then be the near-horizon form of Hawking radiation.

An observer moving with fixed ρ traces out a hyperbola in Minkowski space, therefore this type of motion is called hyperbolic motion.

An observer moving along a path of constant ρ is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function of σ. These modes are constantly Doppler shifted relative to ordinary Minkowski time as the detector accelerates, and they change in frequency by enormous factors, even after only a short proper time.

Translation in σ is a symmetry of Minkowski space: It is a boost around the origin. For a detector coupled to modes with a definite frequency in σ, the boost operator is then the Hamiltonian. In the Euclidean field theory, these boosts analytically continue to rotations, and the rotations close after 2π. So

e2πiH=1.{\displaystyle e^{2\pi iH}=1.}

The path integral for this Hamiltonian is closed with period 2π which guarantees that the H modes are thermally occupied with temperature 1/2π. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle σ, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in σ at position ρ has a frequency which is determined by the square root of the (absolute value of the) metric at ρ, the redshift factor. From the equation for the line element given above, it is easily seen that this is just ρ. The actual inverse temperature at this point is therefore

β=2πρ.{\displaystyle \beta =2\pi \rho .}

Since the acceleration of a trajectory at constant ρ is equal to 1/a, the actual inverse temperature observed is

β=2πa.{\displaystyle \beta ={\frac {2\pi }{a}}.}

Restoring units yields

kBT=ℏa2πc.{\displaystyle k_{\text{B}}T={\frac {\hbar a}{2\pi c}}.}

The temperature of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration of g = 7000981000000000000♠9.81 m·s−2, is only 6980399999999999999♠4×10−20 K. For an experimental test of the Unruh effect it is planned to use accelerations up to 7026100000000000000♠1026 m·s−2, which would give a temperature of about 7005400000000000000♠400000 K.[8][9]

The Rindler derivation of the Unruh effect is unsatisfactory to some, since the detector's path is super-deterministic. Unruh later developed the Unruh–DeWitt particle detector model to circumvent this objection.

The Unruh effect would also cause the decay rate of accelerating particles to differ from inertial particles. Stable particles like the electron could have nonzero transition rates to higher mass states when accelerating at a high enough rate.[10][11][12]

Although Unruh's prediction that an accelerating detector would see a thermal bath is not controversial, the interpretation of the transitions in the detector in the non-accelerating frame is. It is widely, although not universally, believed that each transition in the detector is accompanied by the emission of a particle, and that this particle will propagate to infinity and be seen as Unruh radiation.

The existence of Unruh radiation is not universally accepted. Smolyaninov claims that it has already been observed,[13] while O'Connell and Ford claim that it is not emitted at all.[14] While these skeptics accept that an accelerating object thermalizes at the Unruh temperature, they do not believe that this leads to the emission of photons, arguing that the emission and absorption rates of the accelerating particle are balanced.